Harmonic maps, pluriharmonic maps and existence of non-trivial parallel 2-forms (Q1385229)

The authors obtain some rigidity results, extending the results of \textit{Y.-T. Siu} [Ann. Math., II. Ser. 112, 73-111 (1980; Zbl 0517.53058)]. Let \(M\) be a compact Riemannian manifold of even dimension \(m>2\), carrying a parallel non-null \(2\)-form \(\alpha\), and let \(N\) be a Kähler manifold with strongly negative curvature tensor field. Then the existence of a harmonic map \(\phi \) from \(M\) to \(N\), of rank at least \(m-1\), implies that \(\alpha \) is a Kähler form on \(M\) and \(\phi \) is holomorphic or antiholomorphic for the complex structure induced on \(M\) by \(\alpha \). If a compact manifold \(M\), admitting a non-null parallel \(2\)-form \(\alpha\), has the homotopy type of a compact Kähler manifold with strongly negative curvature tensor field, then \(M\) is Kählerian and the manifolds are holomorphically or antiholomorphically diffeomorphic. The authors obtain an upper bound for the rank of harmonic maps in terms of the rank of the \(2\)-form \(\alpha \) and deduce some vanishing theorems.